A card is selected from a deck of 52 playing cards. Find the probability of selecting


· a prime number under 10 given the card is a heart. (1 is not prime.)
· a diamond or heart given the card is red.
· a King, given that the card is not a spade.

Show step by step work. Give all solutions exactly in reduced fraction form

conditional probability:

P(A|B) read: prob of A, given B
= P(A∩B)/P(B)

let A be all primes under 10
B is all hearts
so A∩B is 2H, 2H, 5H, 7H, there are 4 of those

prob(prime under 10 | red) = (4/52)/(1/4) = 4/13

Here is a page with a nice explanation and examples of conditional probability.

(Broken Link Removed)

To find the probability of selecting a particular card from a deck, we need to determine the number of favorable outcomes (cards that match the given conditions) and divide it by the total number of possible outcomes (total number of cards in the deck).

1. Probability of selecting a prime number under 10 given the card is a heart:

There are 13 hearts in a deck of 52 cards, and to find the prime numbers under 10, we consider the numbers 2, 3, 5, and 7. Out of these four numbers, only one of them is a heart, which is the number 2. Therefore, the number of favorable outcomes is 1.

The total number of possible outcomes (heart cards) is 13, as there are 13 hearts in a deck.

So, the probability of selecting a prime number under 10 given the card is a heart is:

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 1 / 13

The solution in simplified fraction form is 1/13.

2. Probability of selecting a diamond or heart given the card is red:

There are 26 red cards in a deck of 52 cards (13 diamonds and 13 hearts). To determine the number of favorable outcomes (cards that are either a diamond or a heart), we count the number of diamonds and hearts, which is 13 + 13 = 26.

The total number of possible outcomes is still 26, as we are considering only the red cards.

So, the probability of selecting a diamond or heart given the card is red is:

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 26 / 26

The solution in simplified fraction form is 1/1, which means the probability is 1.

3. Probability of selecting a King given that the card is not a spade:

There are 52 cards in a deck, and we know that there are 4 Kings in total (one for each suit - hearts, diamonds, clubs, and spades). However, since we are excluding the spades, we only need to consider the Kings of the other three suits.

So, the number of favorable outcomes (Kings other than spades) is 3.

The total number of possible outcomes is 52, as we are considering all the cards in the deck.

Therefore, the probability of selecting a King given that the card is not a spade is:

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 3 / 52

The solution in simplified fraction form is 3/52.

Thus, the probabilities are:
1. Probability of selecting a prime number under 10 given the card is a heart: 1/13
2. Probability of selecting a diamond or heart given the card is red: 1/1
3. Probability of selecting a King given that the card is not a spade: 3/52